#139006

order	512
nilpotency class	4
isoclinism	isoclinic class of #138768
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	#2408	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×D₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

685 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 685 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 16 points:

(1,2,5,8,6,3,4,7)(9,10)(11,12,14,16)(13,15);(1,3,6,2)(4,7,5,8)(9,11,16,15,13,14,12,10);(1,2,6,3)(4,8,5,7)(9,12,13,16)(10,14,15,11)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=gh,d^2=ghi,e^2=ghi,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,h^b=hi,d^c=dgh,e^c=eghi,f^c=fgh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46202);